Integrable deformations, bi-Hamiltonian structures and nonintegrability of a generalized Rikitake system

Huang, Kaiyin; Shi, Shaoyun; Xu, Zhiguo

doi:10.1142/s0219887819500592

Cited by 10 publications

(7 citation statements)

References 34 publications

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“…In particular, the existence of first integrals plays a crucial role in the integrability of ordinary differential equations [17][18][19][20]. If the considered system of ordinary differential equations admits a straight line solution, by the differential Galois method, one can usually prove that this system has no rational first integrals for almost all the parameters [21][22][23]. However, this method cannot tell whether this system is integrable for the remaining parameters.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

On the Integrability of the SIR Epidemic Model with Vital Dynamics

Ding

2020

Advances in Mathematical Physics

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In this paper, we study the SIR epidemic model with vital dynamics Ṡ=−βSI+μN−S,İ=βSI−γ+μI,Ṙ=γI−μR, from the point of view of integrability. In the case of the death/birth rate μ=0, the SIR model is integrable, and we provide its general solutions by implicit functions, two Lax formulations and infinitely many Hamilton-Poisson realizations. In the case of μ≠0, we prove that the SIR model has no polynomial or proper rational first integrals by studying the invariant algebraic surfaces. Moreover, although the SIR model with μ≠0 is not integrable and we cannot get its exact solution, based on the existence of an invariant algebraic surface, we give the global dynamics of the SIR model with μ≠0.

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

On the Integrability of the SIR Epidemic Model with Vital Dynamics

Ding

2020

Advances in Mathematical Physics

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“…In [4], altering the constants of motion, integrable deformations of the Euler top were constructed. In the same manner, in [5][6][7][8][9], integrable deformations of some three-dimensional systems were obtained. Moreover, in [10], the integrable deformations method for a three-dimensional system of differential equations was presented.…”

Section: Introductionmentioning

confidence: 90%

On the Integrable Deformations of the Maximally Superintegrable Systems

Lăzureanu

2021

Symmetry

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In this paper, we present the integrable deformations method for a maximally superintegrable system. We alter the constants of motion, and using these new functions, we construct a new system which is an integrable deformation of the initial system. In this manner, new maximally superintegrable systems are obtained. We also consider the particular case of Hamiltonian mechanical systems. In addition, we use this method to construct some deformations of an arbitrary system of first-order autonomous differential equations.

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“…q are linearly independent almost everywhere and commute with each other, that is, This definition is introduced by Bogoyavlenskij [25] and is considered as a generalization of the famous Liouville integrability for Hamiltonian systems. Recently, using the differential Galois theory, many scholars have studied non-integrability in Bogoyavlenskij sense of models that appear in physics, economics, biology, chemistry and others [26], [27], [28], [29], [30].…”

Section: Theorem 23 (Existence Of Hopf Bifurcation) Assume That (Hmentioning

confidence: 99%

“…Given initial value (z 1 , z 2 ) = (z 1 (0), z 2 (0)), the solution of system (27) is, the solution of system (27) is…”

Section: Theorem 23 (Existence Of Hopf Bifurcation) Assume That (Hmentioning

confidence: 99%

Complex dynamics in a quasi-periodic plasma perturbations model

Zhang¹,

Yang²

2021

DCDS-B

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In this paper, the complex dynamics of a quasi-periodic plasma perturbations (QPP) model, which governs the interplay between a driver associated with pressure gradient and relaxation of instability due to magnetic field perturbations in Tokamaks, are studied. The model consists of three coupled ordinary differential equations (ODEs) and contains three parameters. This paper consists of three parts: (1) We study the stability and bifurcations of the QPP model, which gives the theoretical interpretation of various types of oscillations observed in [Phys. Plasmas, 18(2011):1-7]. In particular, assuming that there exists a finite time lag τ between the plasma pressure gradient and the speed of the magnetic field, we also study the delay effect in the QPP model from the point of view of Hopf bifurcation. (2) We provide some numerical indices for identifying chaotic properties of the QPP system, which shows that the QPP model has chaotic behaviors for a wide range of parameters. Then we prove that the QPP model is not rationally integrable in an extended Liouville sense for almost all parameter values, which may help us distinguish values of parameters for which the QPP model is integrable. (3) To understand the asymptotic behavior of the orbits for the QPP model, we also provide a complete description of its dynamical behavior at infinity by the Poincaré compactification method. Our results show that the input power h and the relaxation of the instability δ do not affect the global dynamics at infinity of the QPP model and the heat diffusion coefficient η just yield quantitative, but not qualitative changes for the global dynamics at infinity of the QPP model.

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Integrable deformations, bi-Hamiltonian structures and nonintegrability of a generalized Rikitake system

Cited by 10 publications

References 34 publications

On the Integrability of the SIR Epidemic Model with Vital Dynamics

On the Integrability of the SIR Epidemic Model with Vital Dynamics

On the Integrable Deformations of the Maximally Superintegrable Systems

Complex dynamics in a quasi-periodic plasma perturbations model

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